Isaac Newton explicitly constructed four-point parabolas in Arithmetica Universalis (1707). In this Demonstration, you can drag four points that define two parabolas most of the time. By Sylvester's four-point theorem, the points form a concave set with odds 25/36. The corners of a rectangle also cannot be on a parabola. The precise odds that four random points can define a parabola appears to be an unsolved problem.
If the pair of parabolas does exist, the foci are shown as extra points of the same color. Otherwise, in the case of one point being inside the other three, the interior point is colored red.